Zero-Momentum Cyclotron Spin-Flip Mode in a Spin-Unpolarized Quantum Hall System

0.16 MB·

by  S. Dickmann

Zero-Momentum Cyclotron Spin-Flip Mode in a
Spin-Unpolarized Quantum Hall System
S. Dickmann, and I.V. Kukushkin
5
0
Institute for Solid State Physics, Russian Academy of Scieces, Chernogolovka, 142432 Russia
0
2

n We report on a study of the zero-momentum cyclotron spin-flip excitation in the
a
J =2 quantum Hall regime. Using the excitonic representation the excitation energy
V
0
2 is calculated up to the second order Coulomb corrections. A considerable negative

] exchange shift relative to the cyclotron gap is established for cyclotron spin-flip
l
l
a
excitations in the the spin-unpolarized electronic system. Under these conditions
h
-
s this type of states presents the lowest-energy excitations. For a fixed filling factor
e
m
( =2) the energy shift is independent of the magnetic field which is in agreement
. V
t
a with recent experimental observations.
m
- PACS numbers: 73.43.Lp, 78.66.Fd
d
n
o
c
[

3 1. It is well known that in a translationally invariant two-dimensional electron system
v
7 Kohn’s theorem1 prohibits coupling of a homogeneous external perturbation to collective
0
7 excitations of the electrons. As a result, the energy of cyclotron excitations (CE) at zero-
1
1 momentum has no contribution from Coulomb interaction and the dispersion of CE starts
4
0 from the cyclotron gap. In addition to inter-Landau-levels cyclotron excitations [magneto-
/
t
a plasma (MP) mode] there are two other branches of collective excitations in the system of
m
- 2D-electrons: intra-Landau levels spin-flip (SF) excitations (spin-waves) and inter-Landau-
d
n levels combined cyclotron spin-flip excitations (CSFE’s). In the case of SF excitations, there
o
c exists Larmor’s theorem which forbids any contribution from Coulomb interaction to the
:
v
i excitation energy at zero-momentum in spin rotationally invariant systems (see, e.g., Ref.
X
r 2). However, in contrast to the CE and SF excitations, there are no symmetry reasons for
a
the absence of many-body corrections to the zero-momentum energy of CSFE’s. Moreover,
it is well established now both theoretically and experimentally3 that for the spin-polarized
electron system ( = 1) the energy of cyclotron spin-flip excitations is strongly shifted to
V
higher values relative to the cyclotron gap due to the exchange interaction. Therefore, the
energy of combined cyclotron spin-flip excitations is a very convenient tool to probe many-
body effects, for example, in the inelastic light scattering measurements performed at zero
momentum. The sensitivity of CSFE energy at q = 0 to many-body effects strongly de-
pends on the spin polarization of the electron system. For the spin-unpolarized electron
2
system ( =2), theory2 developed within the first order perturbation approach in terms of
V
the parameter r = E /h¯ω (E is the characteristic Coulomb energy, ω is the cyclotron
c C c C c
frequency) predicts a zero many-body contribution to the zero-momentum energy of CSFE.
This result is in contradiction with recent experimental data.4 We show below that calcula-
tion of the CSFE zero-momentum energy of for the =2 system performed to within the
V
second order Coulomb corrections yields a considerable negative exchange shift relative to
the cyclotron gap.
The studied system is characterized by exact quantum numbers S, S and q and by a
z
non-exact but ‘good’ quantum number δn corresponding to the change of the single-electron
energy h¯ω δn with an excitation. The relevant excitations with q=0 and δn=1 may be pre-
c
sented in the form Kˆ† 0 , where 0 is the ground state and Kˆ† are “raising” operators:
S,Sz| i | i S,Sz
Kˆ0†,0= npσ√n+1c†n+1,p,σcn,p,σ, Kˆ1†,0= npσ√n+1(−1)σc†n+1,p,σcn,p,σ and Kˆ1†,+/ =
−
P P
np√n+1c†n+1,p, / cn,p, / , [cn,p,σ is the Fermi annihilation operator corresponding to the
↑ ↓ ↓ ↑
P
Landau-gauge state (n,p) and spin index σ = , ]. The commutators with the kinetic-
↑ ↓
energy operator Hˆ1 are [Hˆ1,KˆS†,Sz]≡h¯ωcKˆS†,Sz. (The total Hamiltonian is Hˆtot =Hˆ1+Hˆint,
where Hˆ is the exact Coulomb-interaction Hamiltonian.) If 0 is unpolarized, we
int
| i
have Sˆ2KˆS†,Sz|0i≡S(S+1)KˆS†,Sz|0i, SˆzKˆS†,Sz|0i≡SzKˆS†,Sz|0i and besides get the identity
h0|KˆS,Sz[Hˆint,KˆS†,Sz]|0i≡0 (|0i, to describe the zero’th order ground state). The latter deter-
mines the first-order Coulomb corrections vanishing both for the S=0 MP mode and for the
S =1 triplet states corresponding to the combined spin-cyclotron excitation. At the same
time [Hˆint,Kˆ0†,0]≡01 but [Hˆint,Kˆ1†,Sz] 6=0 which means that the MP mode indeed has no
exchange energy calculated to any order in r , whereas the triplet states have the exchange
c
correction even in terms of r2.
c
The second-order correction, ∆E h¯ω r2, does not depend on the magnetic field since
SF∼ c c
E = αe2/εl . The renormalization factor, α, is determined by the size-quantized wave
C B
function of electrons confined to the quantum well (QW). In the ideal 2D case α = 1.
However, in experiments with comparatively wide QW’s we expect a well reduced value of
α. Our analytical calculation of the second order correction to the CSFE energy is performed
in terms of r assumed to be small.
c
All three triplet states have certainly the same exchange energy, and it is sufficient to
calculate this, e.g., for the CSFE with S = 1 and S = 1. The obtained result confirms
z
−
experimental observations.
2. The most adequate approach to the integer-quantum-Hall calculations is based on the
Excitonic Representation (ER)5,6 technique (see also, e.g., Refs. 7). The latter means that
instead of single-electron states belonging to a continuously degenerate Landau level (LL)
3
we employ the exciton states Q†abq|0i as the basis set. Here |0i is the ground state found in
the zero approximation in r (it remains also the same even calculated within the framework
c
of the mean field approach). The exciton creation operator is defined as5,6,7,8
1
Q†abq = Nφ Xp e−iqxpb†p+q2y ap−q2y . (1)
p
N = A/2πl2 stands for the number of magnetic flux quanta, q=(q ,q ) is the 2D wave
φ B x y
vector in units of 1/l . Binary indexes a and b present both the LL number and spin index.
B
[I.e. a = (n ,σ ), and a in Eq. (1) stands for the corresponding annihilation operator;
a a p
when exploiting below the notation a = n or a = n as sublevel indexes, this means that
a=(n,↑) or a=(n,↓), respectively.] The annihilation exciton operator is Qabq≡Q†ba q. The
−
commutation rules define a special Lie algebra:5,7,8
hQ†cdq1,Q†abq2i≡Nφ−1/2he−i(q1×q2)z/2δb,cQ†adq1+q2 (2)
−ei(q1×q2)z/2δa,dQ†cbq1+q2 ,
i
where δ = δ δ is the Kronecker symbol. In the = 2 case we get the following
a,b na,nb σa,σb V
1/2
identity: Nφ− Q†aaq|0i≡δq,0 δa,0+δa,0 |0i.
(cid:0) (cid:1)
The advantage of the exciton states lies in the fact that an essential part of the Coulomb
interaction Hamiltonian may be diagonalized in this basis. In the perturbative approach
the excitonically diagonalized part Hˆ should be included into the unperturbed Hamilto-
ED
nian Hˆ = Hˆ +Hˆ and only the off-diagonal part ˆ = Hˆ Hˆ is considered as a
0 1 ED int int ED
H −
perturbation.5 In the excitonic basis the LL degeneracy becomes well lifted because now
there are Coulomb corrections (depending on the q modulus) to the energies of the basis
states. It is useful to take into account that all terms of the relevant ˆ part may be
int
H
presented in the form (cf. Ref. 5)
e2
Hˆint = 2εlB q,a,b,c,dV(q)hhnanb(q)δσa,σbQ†abqi (3)
P
× hncnd(−q)δσc,σdQ†cd q .
h − i
Here 2πV(q) is the dimensionless 2D Fourier component of the averaged Coulomb potential
(in the ideal 2D case V =1/q), and h (q) = (k!/n!)1/2e q2/4(q )n kLn k(q2/2) are the ER
kn − − k−
−
“building-block” functions (Ln is the Laguerre polynomial, q = i (q iq ); cf. also Ref.
k ± ∓√2 x± y
2 and Refs. 5,7). The functions h satisfy the identity: h (q) h ( q).
kn ∗kn ≡ nk −
At the = 2 filling the CSFE state calculated within the zero order in ˆ is sim-
int
V H
ply |SFi = Qˆ†01|0i (the notation Qˆ†ab0 = Qˆ†ab is employed). We thus have HˆEDQ†01|0i =
h0|QabqHˆintQ†01|0i=0 for any indexes q and ab (one could check it directly using the ER
approach; see also Ref. 2). Action of ˆ on the SF state leads to two- and or even
int
H | i
three-exciton states. Therefore the excitonic basis should be extended.
4
3. In principle, there are eight different kinds of possible two-exciton states at = 2.
V
In our case the relevant ones are those corresponding to spin numbers S = 1 and S=0,
z
−
namely: |ν,1i = Q†0n2−qνQ†0n1qν|0i, |ν,2i = Q†0n2−qνQ†0n1qν|0i, |ν,3i = 21Q†0n2−qνQ†0n1qν|0i,
|ν,4i = Q†0n2−qνQ0†n1qν|0i, and |ν,5i = 21Q†0n2−qνQ†0n1qν|0i (certainly only the states with
zero total momentum should be considered). We have used here ν as a composite index
correspondingtotheset (n ,n ,q ). Thetwo-excitonstatesofdifferent typesareorthogonal,
1 2 ν
i.e. I,ν µ,J =0ifI=J [µistheset (m ,m ,q ), belowλ = (l ,l ,q ),...]. However, within
1 2 µ 1 2 λ
h | i 6
the same type their orthogonalization rules should be defined in a special way.
First, let us consider a combination
f ν,I (4)
ν
| i
Xν
(summation is performed over all components of the composite index). In this case the
function f = f(n ,n ,q ) formally turns out to be non-uniquely defined because only a
ν 1 2 ν
certain transform of this has a physical meaning. Indeed, actually only a projection of the
sum (4) onto a certain two-exciton state µ,J would be of any sense. With the help of
| i
commutation rules we obtain
f J,µ ν,I δ f , (5)
ν I,J µ I
h | i ≡ { }
Xν
(cf. Ref. 5). Here the curly brackets mean the transform
f =f N 1 (I)f , if I = 1, 2 or 4;
{ ν}I ν− φ− λFνλ λ (6)
and f =1 fP N 1 (I)f , if I = 3 or 5.
{ ν}I 2 ν− φ− λFνλ λ
(cid:16) (cid:17)
P
(I)
The definition of the kernels f is also parametrized by the kind I of the state, namely:
Fνλ λ
Fν(1λ)=Fν(3λ)=Fν(5λ)≡δn1,l1δn2,l2ei(qν×qλ)z, (7)
Fν(2λ)≡δn1,l2δn2,l1e−i(qν×qλ)z, and Fν(4λ)≡0.
Note that the transform ... is to within a factor equivalent to its double application:
I
{ }
f =K f , where K =K =2 and K =K =K =1. Therefore, if we replace, e.g.,
I I I I 1 2 3 4 5
{{ } } { }
f f +K ϕ ϕ (ϕ is an arbitrary function), then this operation does not affect
ν ν I ν λ I ν
→ −{ }
the combinations (4) and (5). So, only the “antisymmetrized” part f contributes to the
ν I
{ }
matrix-element calculations. The origin of this feature of the two-exciton states is related
to the permutation antisymmetry of the total wave function describing the electron system
studied (cf., e.g., Refs. 5,9). There is also a useful identity
w(n1,n2)fν∗{gν}I≡ w(n1,n2){fν}∗I gν, (8)
Xν Xν
which is valid for any kinds of the transforms ... if the function w in Eq. (5) is assumed
I
{ }
to be such that w(n ,n ) w(n ,n ). In particular, Eq. (5) gives the equations:
1 2 2 1
≡
I,ν µ,J δ δ(I) , (9)
h | i≡ I,J{ νµ}I
5
where
(1) (2) (4)
δ =δ =δ δ δ δ and
νµ νµ νµ ≡ n1,m1 n2,m2 qν,qµ
δ(3)=δ(5) 1 δ δ δ +δ δ δ .
νµ νµ ≡2 n1,m1 n2,m2 qν,qµ n1,m2 n2,m1 qν,−qµ
(cid:0) (cid:1)
(I) (1) (1)
Summation in the δ transform is performed over the first index: e.g. δ δ
νµ I νµ 1 νµ
{ } { } ≡ −
(1)
/N , and so on.
νµ φ
F
4. The first-order corrections (in terms of ˆ ) to the CSFE energy are presented as an
int
H
expansion over the two-exciton states ν,I and three-exciton states Q† ν,I , namely:
| i 01| i
(I)
|SF,i=Q†01|0i+ I=1,2 ν Cν |ν,Ii (10)
P P (I)
+ I=3,4,5 ν Cν Q†01|ν,Ii.
P P
A regular application of the perturbative approach10 leads to the following expression for
the exchange correction to the energy: ∆ESF=hSF|HˆintQ†01|0i. Substituting |SF,i from Eq.
(10) we see that the contribution of the two-exciton states to the energy arises only due to
the terms of Eq. (3) which do not commute with Q† :
01
∆E1−2= I=1,2 νCν(I)∗hI,ν|hHˆint,Q†01i|0i . (11)
P P
(I)
The coefficients C are determined by the equations
ν
µCµ(I)hI,ν|µ,Ii=−hI,ν|[Hˆint,Q†01]|0i/∆ν (12)
P
(I=1,2), where ∆ =h¯ω (n +n 1)>0 stands for the difference of the cyclotron energies in
ν c 1 2
−
the states ν,I and Q† 0 . Calculating the commutator in Eqs. (11)-(12) [employing the
| i 01| i
rules (2)], and then using the properties (5) and (8) of the summation over index, we obtain
∆E = N 1 F F /(n +n 1) (13)
1−2 − φ− I=1,2 ν { ν}I ν∗ 1 2−
P P
[in units of 2Ry=(e2/εl )2/h¯ω =m e4/ε2h¯2], where
B c ∗e
F =V(q )[h (q ) δ h (q )]h ( q ). (14)
ν ν 1n1 ν − 1,n1 00 ν 0n2 − ν
Now we calculate the contribution ∆E which is determined by the three-exciton states
3 5
−
[see Eq. (10)]. This correction arises from the commuting part (with Q†01) of Hˆint acting on
the state Q† 0 , i.e.
01| i
∆E3−5= Cν(I)∗hI,ν|Q01Q†01Hˆint|0i . (15)
I=X3,4,5Xν
The equations for the coefficients are
(I)
µCµ DI,ν|Q01Q†01|µ,IE (16)
P
=− I,ν|Q01Q†01Hˆint|0 /∆˜ν
D E
6
(I=3,4,5), where ∆˜ν=h¯ωc(n1+n2)≥ 2. Substituting Q01Q†01≡Nφ−1/2(Q00−Q11)+Q†01Q01 into
Eqs. (15)-(16) we deduce that the operator Q†01Q01 gives no contribution, whereas action of
the remaining terms reduces the convolutions in Eqs. (15)-(16) to the “bra-ket” products of
two-exciton states. In so doing we find a huge contribution (eventually N ) into Eq. (15)
φ
∼
1/2
due to the commuting part of Nφ− (Q00−Q11), which is actually nothing else but the second
order correction (in terms of r ) to the ground state, namely: ∆E = C(I) 0 ˆ ν,I .
c 0 ν int
h |H | i
ν;IP=3,4,5
According to Eq. (16)
C(I) = [e2/(εl h¯ω )] G /(n +n ) (17)
ν I − B c { ν}I 1 2
(cid:8) (cid:9)
(I=3,4,5) with
G =V(q )h (q )h ( q ). (18)
ν ν 0n1 ν 0n2 − ν
The non-commuting part determines the corrections to the bra-vectors in Eq. (15). For the
I=3 states we get
1
1/2
Nφ− (cid:20)Q00−Q11, 2Q†0n2 qνQ†0n1qν(cid:21)|0i=−2|ν,3i/Nφ,
−
and correspondingly (1+δ ) ν,4 /N and (δ +δ ) ν,5 /N at I=4 and I=5 [the
− n1,1 | i φ − n1,1 n2,1 | i φ
identities (2) have been used]. The similar corrections to the bra-vectors in Eq. (16) do not
(I)
affect the equation (17) for C .
ν
The desirable exchange shift should be measured from corrected energy of the ground-
state. We keep thus in Eq. (15) only the contribution of the non-commuting part (i.e.
considering ∆E ∆E ∆E ). Then by substituting Eq. (3) for ˆ into Eq. (15)
3 5 3 5 0 int
− → − − H
and using again the summation rules (5) and (8) we find from Eqs. (15) and (17) the I=3 5
−
correction
∆E = 1 [(2+δ +δ ) G
3−5 Nφ ν n1,1 n2,1 { ν}3 (19)
P
+(1+δ )G ]G /(n +n )
n1,1 ν ∗ν 1 2
(in units of 2Ry). The combination with Eq. (13) yields
∆E = ∆E +∆E . (20)
SF 1 2 3 5
− −
The sum over ν in Eqs. (13) end (19) means summation over n and n and the in-
1 2
tegration over q . This is a routine procedure and the suitable sequence of operations
ν
is as follows. First we perform the summation over all of n 1 and n 1 keeping
1 2
≥ ≥
the sum n = n +n fixed. Then we make the integration over q . According to the
ν 1 2 ν
above definition, the transforms F and G already contain an integration, there-
{ ν}I { ν}I
fore some terms in Eq. (8) present twofold integration over 2D vectors q and q . Re-
λ ν
ally the latter, with the help of formula (2π) 2 dq dq U(q ,q )(q q )me i(q1 q2)z
− 1 2 1 2 1+ 2 ± ×
− ≡
RR
0∞ 0∞dq1dq2(q1q2)mU(q1,q2)J m(q1q2)(Jm istheBesselfunction,U isanarbitraryfunction),
±
R R
7
is reduced to integration over absolute values q and q . Finally the numerical summation
λ ν
over n is performed.
ν
In so doing, a simplifying circumstance was found: all of the twofold-integration terms
cancel each other in the final combination (20). (This feature is not a general one but only
inherent in our specific case.11) All the rest terms result in the following expression:
∆E = ∞ S 1 21−n, where
SF −n=2 nn(−n2−1) (21)
P
Sn=n2! 0∞dqq2n+3V2(q)e−q2.
R
For the ideally 2D system we have S 1, and the summation may be easily performed,
n
≡
yielding ∆E = (ln2 1)/2 = 0.1534... (in units of 2Ry).
SF
− −
5. So, the shift is negative and the exchange interaction lowers thereby the CSFE energy
relative to the singlet MP mode. The sign of the shift presents an expectable result. Indeed,
the second-order correction to the energy of a low-lying excitation should be presumably neg-
ative due to the same reasons which determine the inevitably negative sign of the correction
to the ground state energy. Another remarkable feature of the found shift is its independence
of the magnetic field.
Due to the q=0 condition the studied state is optically active and should be observed in
photo-luminescent and inelastic light scattering (ILS) measurements. In the recent work 4
the ILS was studied in a single 30nm AlGaAs/GaAs QW in the situations where =2;4.
V
The triplet and MP cyclotron excitations are manifested as peaks in the ILS spectra. The
measurements were performed in magnetic fields varied in a wide range, but with the filling
factor kept constant. The central triplet line is shifted downward from the cyclotron energy
by 0.35meV independently of the B magnitude. Thus, a qualitative agreement with our
calculation is obvious.
Quantitative comparison should be done with taking into account of finite thickness of
a two-dimensional electron gas. The calculation in Fig. 1 incorporates the effect of the
finite width of the 2D layer. This is carried out by writing the Coulomb vertex as V(q)=
F(qw)/q, where the form factor F(qw) is parametrized by an effective thickness w. The
latter characterizes the spread of the electron wavefunction in the perpendicular direction.
If the variational envelope function is chosen in the form ψ(z) 2 exp( z2/2w2), then
| | ∼ −
F(qw)=ew2q2erfc(wq) (see Ref. 12). Exactly this form factor is employed in the calculation
based on Eq. (21). Taking into account the value of Ry=5.67meV in GaAs, we find from
Fig. 1 that the agreement with the experiment is obtained at w 0.5l . This is quite
B
≈
reasonable value for the 30nm GaAs quantum structure.
As a concluding remark we notice that the triplet cyclotron excitation in spin-unpolarized
electron system seems to have been observed earlier,13 although in this paper experimental
8
observations were related to the roton minimum and a different experimental dependence of
energy shift on magnetic field was detected.
We acknowledge support by the Russian Fund of Basic Research. S. D. thanks for hospi-
talitytheMaxPlanckInstituteforPhysics ofComplexSystems(Dresden),whereanessential
part of this work has been done.
1 W. Kohn, Phys. Rev., 123, 1242 (1961).
2 C. Kallin and B. I. Halperin, Phys. Rev. B 30, 5655 (1984).
3 A. Pinczuk et al., Phys. Rev. Lett. 68, 3623 (1992).
4 L.V. Kulik, I.V. Kukushkin, S. Dickmann, V.E. Kirpichev, A.B. Van’kov, A.L. Parakhonsky,
J.H. Smet, K. v. Klitzing, and W. Wegscheider, to appear (cond-mat/0501466).
5 S. Dickmann, Phys. Rev. B 65, 195310 (2002).
6 A.B. DzyubenkoandYu.E.Lozovik, Sov.Phys.SolidState25, 874 (1983) [ibid.26, 938 (1984)].
7 S. Dikman, and S. V. Iordanskii, JETP 83, 128 (1996); S. Dickmann and Y. Levinson, Phys.
Rev. B 60 7760 (1999); S. Dickmann, Phys. Rev. B 61, 5461 (2000); and Phys. Rev. Lett. 93,
206804 (2004).
8 Yu.A. Bychkov, and S.V. Iordanskii, Sov. Phys. Solid State 29, 1405 (1987).
9 Yu.A. Bychkov, and E.I. Rashba, JETP 58, 1062 (1983).
10 L.D. Landau and E.M. Lifschitz, Quantum Mechanics (Butterworth-Heinemann, Oxford, 1991).
11 E.g.thesetermsareavailableinthesecondordercorectionstotheskyrmion-antiskyrmiongap[6]
and to the energy of the hole at the =1 filling. We remark that the hole energy correction was
V
calculatedinRef.[6]withouttakingintoaccountthetwofold-integration contribution.Thelatter
constitutes 0.00349 (in the strict 2D limit) and has to be added to the result π2/24+3(ln2)2
− − 4
presented there. Thetotal correction is thereby 0.0544 in units of 2Ry (cf. the numericalresult
−
in Ref. [14]).
12 N.R. Cooper, Phys. Rev. B 55, 1934 (1997).
13 M.A. Eriksson et al., Phys. Rev. Lett. 82, 2163 (1999).
14 S.L. Sondhi et al., Phys. Rev. B 47, 16419 (1993).
9

0.0 0.2 0.4 0.6 0.8 1.0
0.15
0.15
y
R
2
of
s
nit0.10
u
n 0.10
| iSF
E
hift |0.05
S 0.05
0.00
0.0 0.2 0.4 0.6 0.8 1.0 0.00
w / lB
0.0 0.2 0.4 0.6 0.8 1.0
FIG. 1: The CSFE exchange shift is calculated from the formula of Eq. (21) with the modified
Coulomb interaction V(q)=q 1eq2w2erfc(qw); the shift value absolute at w=0 is (1 ln2)/2.
−
−